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Theorem ssmapsn 38403
 Description: A subset 𝐶 of a set exponentiation to a singleton, is its projection 𝐷 exponentiated to the singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ssmapsn.f 𝑓𝐷
ssmapsn.a (𝜑𝐴𝑉)
ssmapsn.c (𝜑𝐶 ⊆ (𝐵𝑚 {𝐴}))
ssmapsn.d 𝐷 = 𝑓𝐶 ran 𝑓
Assertion
Ref Expression
ssmapsn (𝜑𝐶 = (𝐷𝑚 {𝐴}))
Distinct variable groups:   𝐴,𝑓   𝐶,𝑓   𝜑,𝑓
Allowed substitution hints:   𝐵(𝑓)   𝐷(𝑓)   𝑉(𝑓)

Proof of Theorem ssmapsn
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 ssmapsn.c . . . . . . . . 9 (𝜑𝐶 ⊆ (𝐵𝑚 {𝐴}))
21sselda 3568 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
3 elmapi 7765 . . . . . . . 8 (𝑓 ∈ (𝐵𝑚 {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 5959 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovex 6577 . . . . . . . . . . . 12 (𝐵𝑚 {𝐴}) ∈ V
98a1i 11 . . . . . . . . . . 11 (𝜑 → (𝐵𝑚 {𝐴}) ∈ V)
109, 1ssexd 4733 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
11 rnexg 6990 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1211rgen 2906 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1312a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
1410, 13jca 553 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
15 iunexg 7035 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1614, 15syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
177, 16eqeltrd 2688 . . . . . . 7 (𝜑𝐷 ∈ V)
1817adantr 480 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
19 ssiun2 4499 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
2019adantl 481 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
21 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
22 snidg 4153 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2321, 22syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2423adantr 480 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
25 fnfvelrn 6264 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
265, 24, 25syl2anc 691 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2720, 26sseldd 3569 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2827, 6syl6eleqr 2699 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
295, 18, 28elmapsnd 38391 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
3029ex 449 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
3117adantr 480 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐷 ∈ V)
32 snex 4835 . . . . . . . . . 10 {𝐴} ∈ V
3332a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → {𝐴} ∈ V)
34 simpr 476 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
3523adantr 480 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐴 ∈ {𝐴})
3631, 33, 34, 35fvmap 38382 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝐷)
376idi 2 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
38 rneq 5272 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3938cbviunv 4495 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
4037, 39eqtri 2632 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4136, 40syl6eleq 2698 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
42 eliun 4460 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4341, 42sylib 207 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
44 simp3 1056 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
45 simp1l 1078 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4645, 21syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
47 eqid 2610 . . . . . . . . . . 11 {𝐴} = {𝐴}
48 simp1r 1079 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
49 elmapfn 7766 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓 Fn {𝐴})
5048, 49syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
511sselda 3568 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵𝑚 {𝐴}))
52 elmapfn 7766 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐵𝑚 {𝐴}) → 𝑔 Fn {𝐴})
5351, 52syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
54533adant3 1074 . . . . . . . . . . . 12 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
55543adant1r 1311 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5646, 47, 50, 55fsneqrn 38398 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓𝐴) ∈ ran 𝑔))
5744, 56mpbird 246 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
58 simp2 1055 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5957, 58eqeltrd 2688 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
60593exp 1256 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6160rexlimdv 3012 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6243, 61mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓𝐶)
6362ex 449 . . . 4 (𝜑 → (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓𝐶))
6430, 63impbid 201 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
6564alrimiv 1842 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
66 nfcv 2751 . . 3 𝑓𝐶
67 ssmapsn.f . . . 4 𝑓𝐷
68 nfcv 2751 . . . 4 𝑓𝑚
69 nfcv 2751 . . . 4 𝑓{𝐴}
7067, 68, 69nfov 6575 . . 3 𝑓(𝐷𝑚 {𝐴})
7166, 70dfcleqf 38281 . 2 (𝐶 = (𝐷𝑚 {𝐴}) ↔ ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
7265, 71sylibr 223 1 (𝜑𝐶 = (𝐷𝑚 {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∪ ciun 4455  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  vonvolmbllem  39550  vonvolmbl2  39553  vonvol2  39554
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